73
ISSN 1472-2739 (on-line) 1472-2747 (printed)
Algebraic & Geometric Topology
Volume 4 (2004) 73–80
Published: 14 February 2004
ATG
On symplectic fillings
John B Etnyre
Abstract In this note we make several observations concerning symplectic
fillings. In particular we show that a (strongly or weakly) semi-fillable
contact structure is fillable and any filling embeds as a symplectic domain
in a closed symplectic manifold. We also relate properties of the open book
decomposition of a contact manifold to its possible fillings. These results
are also useful in showing the contact Heegaard Floer invariant of a fillable
contact structure does not vanish [28] and property P for knots [18].
AMS Classification 53D05, 53D10; 57M50
Keywords Tight, symplectic filling, convexity
1
Introduction
It is important to consider various notion of “symplectic filling” when studying
contact and symplectic manifolds (especially in low dimensions). In particular,
it is useful to understand when a contact structure on a three manifold has a
symplectic filling, what type of filling is it and if there are restrictions on the
topology of the filling.
Consider a contact three manifold (M, ξ) and symplectic manifold (X, ω) with
M a boundary component of X and ω|ξ > 0. If all the boundary components of X are convex then (X, ω) is a weak (or possibly strong) semi-filling
of (M, ξ). (Throughout this paper all contact structures will be (co)oriented,
contact manifolds will be oriented by their contact structures and symplectic
manifolds will be oriented by their symplectic structures.) We say the boundary component M of X can be symplectically capped off if there is a symplectic
manifold (X 0 , ω 0 ) such that X 0 \ M = X ∪ C, ω 0 |X = ω and C is compact.
(Note this implies that ∂C = −M.) The symplectic manifold (C, ω 0 |C ) is called
a symplectic cap for (M, ξ) ⊂ (X, ω). (The notation A \ B means the union of
the metric completions of the components of A minus B.) Many of our results
rely on the following theorem.
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John B Etnyre
Theorem 1.1 If (X, ω) is a (weak or strong) symplectic filling (or semi-filling)
of (M, ξ) then the boundary component M of X can be capped off symplectically.
We have the following immediate corollaries.
Corollary 1.2 A weakly, respectively strongly, semi-fillable contact structure
is weakly, respectively strongly, fillable.
Corollary 1.3 Any symplectic four manifold with weakly or strongly convex
boundary components can be embedded as a domain in a closed symplectic
manifold.
The theorem and both corollaries have also been established by Eliashberg in
[3]. For strong (semi)fillings these results were essentially proven by Gay in
[13] and follow from Theorem 1.3 in [11]. Moreover, after writing this paper
the author learned that Stipsicz and Ghiggini also know how to prove these
results for strong fillings. Moreover, in [29], Stipsicz has a different proof of
Lemma 3.1 (below), which is the key ingredient in the proof of Theorem 1.1.
In [23] Lisca and Matić and in [1] Akbulut and Ozbagci proved the corollaries for Stein fillings. Not only do these corollaries illuminate the nature of
symplectic and contact geometry in low dimensions they have many important
applications. Specifically, Kronheimer and Mrowka [18] can use Corollary 1.2
to prove the Property P conjecture for knots and Ozsváth and Szabó [28] can
use Corollary 1.3 to shown that their Heegaard Floer contact invariant does not
vanish for fillable contact structures. Moreover, they can use the corollary to
give an alternate proof that if p surgery on a knot K yields a manifold which
is (orientation preserving) homeomorphic to L(p, 1) then K is the unknot (this
result first appeared in [20] using Seiberg–Witten Floer Homology).
We now turn to the topology of a symplectic filling. A fundamental result of
Giroux [14] (see also [16]) says that a contact structure ξ on a three manifold M is always supported by an open book (L, φ). That is, comes from the
construction of Thurston and Winkelnkemper [30]; or more explicitly, there is
a link L in M that is transverse to ξ such that M \ L is fibered by Seifert
surfaces for L and the contact structure can be isotoped to be arbitrarily close
to the fibers in the fibration (while keeping L transverse). The monodromy of
the open book φ is the monodromy of the fibration M \ L.
Theorem 1.4 Suppose the contact structure ξ on a three manifold M is
supported by the open book (L, φ) and there is a diffeomorphism ψ of the
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75
fiber such that (L, ψ ◦ φ) is an open book for S 3 and ψ can be written as
a composition of right handed Dehn twists. If (X, ω) is a, weak or strong,
symplectic semi-filling of (M, ξ) then (X, ω) is a filling of (M, ξ). Moreover, if
M is a rational homology sphere then H1 (X; Z) finite (and trivial if M is an
integral homology sphere) and the intersection form on H2 (X; Z) is negative
definite.
Lisca (in [21, 22]) and Ohta–Ono (in [27]) have obtained a similar result under
the assumption that M admits a metric of positive scalar curvature (but no
assumption on the supporting open book). We note that if φ−1 can be written
as a product of right handed Dehn twists then the conditions of the theorem
are satisfied (see the proof of Lemma 3.1). If φ−1 can be so written then
φ can be written as product of left handed Dehn twists. One might expect
that this implies the associated contact structure is overtwisted, but this is not
necessarily true, only if the open book “destabilizes.” In general it is quite
subtle to decide if an open book destabilizes. It is also difficult to make general
statements about representing a diffeomorphism in terms of Dehn twists.
Theorem 1.5 Under the hypothesis of Theorem 1.4 any symplectic filling of
(M, ξ) can be embedded as a symplectic domain in a rational surface.
While these theorems seem to provide potentially quite useful insights into the
monodromy of open books we content ourselves here with the following simple
applications.
Corollary 1.6 Let φ be any monodromy associated to any tight contact structure coming from perturbing a Reebless foliation into a contact structure as in
[9]. Then φ cannot be expressed as a composition of left handed Dehn twists.
Moreover, there is no diffeomorphism ψ such that φ ◦ ψ is the monodromy for
an open book of S 3 and ψ is a composition of right handed Dehn twists.
As a specific example any monodromy map associated to any tight contact
structure on T 3 cannot be expressed as the composition of left handed Dehn
twists.
Proof As shown in [9], such a contact structure on M is semi-filled by a
symplectic form on M × [0, 1]. This contradicts the conclusion of Theorem 1.4
and thus the contact structure cannot satisfy the hypothesis of Theorem 1.4.
Algebraic & Geometric Topology, Volume 4 (2004)
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John B Etnyre
Acknowledgments I thank Yasha Eliashberg for sharing an early version
of [3] with me which made me aware of the many interesting applications of
symplectic caps. I also thank the referee who provided invaluable suggestions
for improvements on the paper. In addition, I gratefully acknowledge support
from an NSF Career Grant (DMS–0239600) and FRG-0244663.
2
Symplectic fillings
We recall the various notions of symplectic fillings of contact manifolds. (For
more details, see the survey paper [10], and for further discussion of the notions
of fillability see [3].)
A symplectic manifold (X, ω) is said to have strongly convex boundary if there is
a vector field v defined in the neighborhood of ∂X that points transversely out
of X and dilates ω (ie, Lv ω = ω ). The form α = (ιv ω)|∂X is a contact form on
∂X. The contact manifold (M = ∂X, ξ) on the boundary is said to be strongly
symplectically fillable by (X, ω). If (X, ω) supports a Stein structure then (M, ξ)
is said to be holomorphically fillable (this is sometimes called Stein fillable).
A symplectic manifold (X, ω) is said to have weakly convex boundary if ∂X
admits a contact structure ξ such that ω|ξ > 0. A contact manifold (M, ξ) is
weakly symplectically fillable if it is the weakly convex boundary of a symplectic
manifold. A contact structure is said to be weakly (or strongly) semi-fillable if
it is one component of the boundary of a weakly (or strongly) convex symplectic
manifold. (Note a Stein manifold always has a connected boundary so there is
no notion of holomorphically semi-fillable.) When talking about strong or weak
symplectic fillings we will frequently drop the word “symplectic” and just refer
to strong or weak fillings.
The following diagram indicates the hierarchy of contact structures.
Tight
∪6|
Weakly semi-fillable %
||
Weakly fillable
%
Strongly semi-fillable
||
Strongly fillable
∪
Holomorphically fillable
The properness of the inclusion of the set of weakly symplectically semi-fillable
contact structures into the set of tight contact structures was shown in [12]
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On symplectic fillings
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(for further examples see [24]). The properness of the inclusion of the set of
strongly fillable structures into the set of weakly fillable structures is due to
Eliashberg [4] (for further examples see [2]). The equalities of the semi-fillings
and “honest” fillings is the content of Corollary 1.2 and [3]. Though there are
strong fillings of a contact structure that are not holomorphic fillings (because
there is more than one boundary component [25]) it is not currently known if
strongly fillable contact structures are also holomorphically fillable.
If M is a rational homology sphere, a weak symplectic filling can be modified
into a strong symplectic filling [27] (the germ of the argument originally appeared in [6], see also [3]). If M has a positive scalar curvature metric, then
moreover a semi-filling is automatically a one boundary component filling, and
all four notions of symplectic filling become the same (this is due to Lisca [21]
and Ohta–Ono [27]). The proof builds on the work of Kronheimer and Mrowka
[19] and relies on Seiberg–Witten theory.
3
Proof of Theorem 1.1
For the proof of Theorem 1.1 we will need the following lemma (see also [29]).
Lemma 3.1 Under the hypothesis of Theorem 1.1, 2–handles may be attached
to X to form a new symplectic manifold with weakly convex boundary (X 0 , ω 0 )
such that X 0 \ M = X ∪ B with ∂B = M 0 ∪ (−M ) where M 0 is a homology
sphere.
With this lemma in hand we prove Theorem 1.1
Proof of Theorem 1.1 Given (M, ξ) and (X, ω) as in the theorem, apply
the above lemma to obtain (X 0 , ω 0 ). Since M 0 is a homology sphere the results
of Ohta–Ono mentioned in the previous section imply ω 0 may be isotoped near
M 0 so that (X 0 , ω 0 ) is strongly convex along M 0 . Theorem 1.3 from [11] says
that any contact manifold has (infinitely many) strongly concave fillings. Since
a strongly convex and strongly concave filling may be glued to form a symplectic
manifold [10] we see that (X 0 , ω 0 ) may be symplectically capped off. Let C be
the cap. Then C ∪ B (B from Lemma 3.1) is the symplectic cap for (M, ξ).
We are left to prove the lemma.
Algebraic & Geometric Topology, Volume 4 (2004)
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John B Etnyre
Proof of Lemma 3.1 Giroux’s result about open books and contact structures implies we can assume that (M, ξ) is supported by an open book with
connected binding (this can be achieved by “positive Hopf stabilization”). Let
φ be the monodromy of the open book. Since the mapping class group is generated by right handed Dehn twists about arbitrary loops plus arbitrary Dehn
twists parallel to the boundary, we can write φ−1 as φ0 ◦ φ1 , where φ0 is a
composition of right handed Dehn twists and φ1 is some number of left handed
Dehn twists parallel to the boundary (we can put all the boundary parallel
twists at the end since they commute with the other Dehn twists). Thus by
attaching 2–handles to X along M in a symplectic way (see [7, 15, 31]) we
can get a symplectic manifold with a boundary component (M 00 , ξ 00 ), such that
(M 00 , ξ 00 ) is supported by an open book with monodromy φ ◦ φ0 which is, say, k
right handed Dehn twists parallel to the boundary. Topologically the manifold
is shown in the Figure 1. Now for each 1–handle in the figure add a 2–handle
−1 on each 2–handle
k 2–handles
− k1
Figure 1: On the left hand side we show an open book with k right handed Dehn twists
parallel to the boundary. The right hand side is homeomorphic to the left hand side.
that runs over it once (this can be done in a symplectic way). We now have a
symplectic manifold with boundary (M 0 , ξ 0 ). Topologically M 0 is obtained by
− k1 surgery on a knot in the three sphere, thus it is a homology sphere.
4
Topology of symplectic fillings
Proof of Theorem 1.4 and 1.5 Given the hypothesized diffeomorphism ψ
we can repeat the proof of Lemma 3.1 and attach 2–handles to X along M
to form a new symplectic manifold (X 0 , ω 0 ) with new boundary component
Algebraic & Geometric Topology, Volume 4 (2004)
On symplectic fillings
79
(S 3 , ξ 0 ). Since ξ 0 is fillable we know it is tight. Thus ξ 0 is the unique tight
contact structure on S 3 (see [5]). McDuff’s well known extension [26] of Gromov’s theorem [17] says that (X 0 , ω 0 ) is symplectomorphic to the blow up of the
standard symplectic structure on B 4 ⊂ C 2 . This of course implies our original
symplectic filling (X, ω) only had one boundary component and that we may
cap off (X 0 , ω 0 ) by a symplectic disk bundle over S 2 to obtain a rational surface.
This completes the proof of Theorem 1.5. To finish the proof of Theorem 1.4
simply note that if a rational homology sphere splits a rational surface as ours
does then the component (X 0 , ω 0 ) must be negative definite and H1 (X; Z) must
be torsional (and trivial if M is an integral homology sphere).
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Department of Mathematics, University of Pennsylvania
209 South 33rd St, Philadelphia, PA 19104-6395, USA
Email: etnyre@math.upenn.edu
URL: http://math.upenn.edu/~etnyre
Received: 7 January 2004
Algebraic & Geometric Topology, Volume 4 (2004)